Problem 49 - Algebra 2 and Trigonometry

Question

In how many ways can 6 transfer students be assigned to seats in a classroom that has 10 empty seats?

Step-by-Step Solution

Answer

There are 151,200 ways to assign the 6 students to 10 seats.

Step 1: Understanding the Problem

We need to assign 6 transfer students to 10 empty seats in a classroom. This is a problem of counting the arrangements, which can be solved using permutations.

Step 2: Choosing Seats for Students

First, we need to choose 6 seats out of the 10 available seats for the students. The number of ways to choose 6 seats from 10 is given by the combination formula: \[ \binom{10}{6} \]

Step 3: Calculating Combinations

Calculate the number of combinations. \[ \binom{10}{6} = \frac{10!}{6!(10-6)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]

Step 4: Assigning Students to Chosen Seats

Once 6 seats are chosen, each seat can be filled by any of the 6 students. Thus, the number of ways to arrange 6 students in 6 seats is given by 6! (factorial of 6). Therefore:\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]

Step 5: Calculating Total Arrangements

Multiply the number of ways to choose the seats by the number of ways to arrange the students in those seats:\[ 210 \times 720 = 151,200 \]

Step 6: Conclusion

The total number of ways the 6 transfer students can be assigned to 10 seats is 151,200.

Key Concepts

Combinations

Combinations allow us to select items from a larger set where the order does not matter. In our case, we need to choose 6 seats out of 10 available, which is a perfect example of combinations. The formula to calculate combinations is: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

• Here, \(n\) is the total number of items to choose from, which is 10 seats.
• \(k\) is the number of items to choose, which is 6 seats.

This formula helps us to count the possible ways to choose the seats. It only considers the selection, not the arrangement. This is why combinations are useful in problems where the sequence doesn't matter.

Factorial

Factorial is a concept often denoted by an exclamation mark \(!\), and it represents the product of an integer and all the integers below it. For instance, the factorial of 6, expressed as 6!, is:\[6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\]Factorials are essential for calculating permutations and combinations, like in the combination formula given above.

• In permutations, factorial helps us determine how to arrange a subset of items.
• In combinations, factorial allows us to calculate how many ways items can be selected.

Understanding factorials is crucial for solving many counting problems in mathematics.

Arrangements

Arrangements are about figuring out how to order or position a subset of items from a larger set once these items are selected. Once we have chosen 6 seats from the 10 available, arranging 6 students in these seats is a matter of permutation, calculated using factorial. Since each seat can be filled by any of the 6 students, we use the factorial of 6 (6!) to find the number of ways to arrange them:\[6! = 720\]

• Each arrangement of students determines a different and unique seating order.
• This highlights the difference between combinations (choosing items) and arrangements (ordering selected items).

Thus, using arrangements, we ensure every possible seating position of the students is counted.

Counting Problems

Counting problems are mathematical queries that require us to find the number of ways certain events can occur under a specific set of conditions. They are fundamental in understanding the breadth of possibilities in different scenarios.

• Many counting problems use permutations and combinations as tools to arrive at the solution.
• In our case, we first use combinations to select the seats and permutations for arranging students.

The tools of counting problems help us arrange items or participants efficiently and are used heavily in statistics, probability, and combinatorics. By breaking the problem into selecting and arranging, we manage complex queries more effectively, making challenges easier to solve.